The Massachusetts Comprehensive Assessment System (MCAS) Release of 2002 Test Items Mathematics Grade 8

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1 The Massachusetts Comprehensive Assessment System (MCAS) Release of 2002 Test Items Mathematics Grade 8 Massachusetts Comprehensive Assessment System: 1

2 Massachusetts Comprehensive Assessment System: 2

3 Session 1, Open Response Question #9 9 A worker placed white tiles around black tiles in the pattern shown in the three figures below. a. Based on this pattern, how many white tiles would be needed for 4 black tiles? b. Based on this pattern, how many white tiles would be needed for 50 black tiles? c. Make a scatterplot of the first five figures in this pattern showing the relationship between the number of white tiles and the number of black tiles. Be sure to label the axes. d. Based on this pattern, explain how you could find the number of white tiles needed for any number, n, of black tiles. Show or explain your work. Reporting Category for item 9: Patterns, Relations, and Algebra Massachusetts Comprehensive Assessment System: 3

4 Question 9 Scoring Guide Score 4 3 Description The response shows a comprehensive understanding of how to extend, represent and generalize a pattern with graphs and symbolic expressions. The response shows a general understanding of how to extend, represent and generalize a pattern with graphs and symbolic expressions. 2 The response shows a basic understanding of patterns. 1 The response shows a minimal understanding of patterns. 0 The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No Response. Score Point 4 a. For a pattern with 4 black tiles you would need 14 white tiles b. For a pattern with 50 black tiles you would need 106 white tiles. c. *See next page. d. To find the number of white tiles for any number you could use the formula below. 2n + 6 2(1)+6 2(2)+6 2(3) Massachusetts Comprehensive Assessment System: 4

5 Score Point #4 continued x = black tiles y = white tiles c. 16Y Score Point 3 A. 14 white tiles would be needed for four black tiles. B. 106 white tiles would be needed to make 50 black tiles. 2b + b C D. n = 2b + b B = black, n = white Massachusetts Comprehensive Assessment System: 5

6 Score Point #3D continued It took a lot of thought to find this formula. I looked at the white and black tiles closely for quite a while. In all the patterns, there was 12 times the number of black tiles, plus 6. This formula works for every pattern of black and white tiles. Score Point 2 A= In my opinion, 6 white tiles would be needed for 4 black tiles. B= There would have to be 100 white blocks. D= In order to find the letter n, you have to add the number of black tiles to 7. And everytime you add another black block add 1 more to 7. Then you have to add 2 to the 7. D= W=B+7+2 W=3+7+2 C= 16 12=3+7+2 also 14 D= W=B+8+2 W= =4+8+2 Everytime you 10 add another black block 8 you have to add another 6 to the black tiles Massachusetts Comprehensive Assessment System: 6

7 Score Point 1 a. 12 white tiles would be needed for four black tiles. b. 150 white tiles would be needed for 50 black tiles. c. I. II 8 white tiles 10 white tiles 1 black tile 12 black tiles III. IV. 12 w. tiles 14 w. tiles 3 b. tiles 4 b. tiles IIV. 16 w. tiles 5 b. tiles d. I could find out the # of tiles needed for every 1 black tile easily. For every one tile there are two other white tiles. For the just one you need to surround it with tiles. Then everytime you add a black one you add two white ones. Massachusetts Comprehensive Assessment System: 7

8 Score Point 0 A. 20 white tiles would be needed for 4 black tiles. B. 230 white tiles would be needed for 50 black tiles. C. White tiles Black tiles D. I added all the tiles up or counted all the white squares surrounding the 4 black ones and added 50 to it than x s it by 4. the 4 stands for the 4 black squares. Massachusetts Comprehensive Assessment System: 8

9 Session 1, Open Response Question #22 22 Lionel and Tracy are playing a game using two six-sided number cubes. The faces of each cube are numbered as shown below. Lionel has a red cube and Tracy has a green cube. To play the game they both roll their cubes at the same time. The numbers that show face up when the cubes stop rolling are used to make a fraction. The number on the red cube is used for the numerator and the number on the green cube is used for the denominator. For example, the results shown below would make the fraction 1 2. Lionel wins 1 point if the fraction formed has a value less than one. Tracy wins 1 point if the fraction has a value greater than one. No one gets a point if the fraction is equal to one. Massachusetts Comprehensive Assessment System: 9

10 Question #22 continued a. Make a list or a table in your Student Answer Booklet of all of the fractions possible from rolling 1 red and 1 green cube. How many total different fractions are there? b. If Lionel (red cube) rolls a 3, what is the probability that Tracy (green cube) wins 1 point? Show your work or explain how you obtained your answer. c. Using your table, what is the probability of each player winning a point on a given turn? Do you think this game is fair to both players? Show your work or explain how you obtained your answer. Reporting Category for item 22: Data Analysis, Statistics, and Probability Massachusetts Comprehensive Assessment System: 10

11 Question 22 Scoring Guide Score 4 3 Description The response shows a comprehensive understanding of how to find the total number of outcomes for a situation and calculate the probability of simple compound events. The response shows a general understanding of how to find the total number of outcomes for a situation and calculate the probability of simple compound events. 2 The response shows a basic understanding of probability. 1 The response shows a minimal understanding of probability. 0 The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No Response. Score Point L 1 3 L 1 4 L 1 5 L 1 6 L 2 1 T L 2 4 L 2 5 L 2 6 L 3 1 T 3 2 T L 3 5 L 3 6 L 4 1 T 4 2 T 4 3 T L 4 6 L 5 1 T 5 2 T 5 3 T 5 4 T L 6 1 T 6 2 T 6 3 T 6 4 T 6 5 T different Fractions tie = blank T = Tracy wins L = Lionel wins Possibilities Winner T T tie L L L Massachusetts Comprehensive Assessment System: 11

12 Score Point #4 continued 2 6 = Tracy Wins = 12 5 Lionel Wins = 12 = 1 6 It is a fair game. The probabilities are equal. Score Point3 A. for the number 2 there are 6 possible fractions. That is the same as for all the numbers on the dice (there are six). That means there are 36 possible fractions. B. Possible combinations = 1 3 >1 >1 =1 <1 <1 <1 Tracy has a 1 3 rolls a 3. chance to get one point if Lionel C. 15 possible fractions out of 36 are greater than 1. That means 5 12 are >1. I used my table to figure that out. This game is fair because both people can get 5 12 chances to get a point. It would not be fair if I counted as a point. Massachusetts Comprehensive Assessment System: 12

13 Score Point 2 A. 1,1 2,4 4,1 5,4 1,2 2,5 4,2 5,5 1,3 2,6 4,3 5,6 1,4 3,1 4,4 6,1 There are a 1,5 3,2 4,5 6,2 total of 36 1,6 3,3 4,6 6,3 combinations. 2,1 3,4 5,1 6,4 2,2 3,5 5,2 6,5 2,3 3,6 5,3 6,6 B. If Lionel rules a (red) 3 there is a 50/50 chance that Tracy will win the point. This is true because it is half way between getting a six or one. I think this game is fair to both players, although Lionel is more likely to win. He has more chances of winning because there are a few more combinations in his favor. Score Point 1 If some on rolling different colors, than there would be 1 6 of all different fractions. If Lionel rolls a 3, Tracy would roll she would probably get 1 out of 3 chances so it is very likely that she will get a point because she has several chances. The probability of each player winning a point on a given turn is 1 out of 6 chances. I think it is not fair to both players because even though you have a couple of chanced you still have 1 out of 6. Since there is two people playing then it would be 1 out of 6 means that only one person can win because, they can t tie. Massachusetts Comprehensive Assessment System: 13

14 Score Point 0 (A) (B) (C) Yes I do why shoulnt they the end bye bye 2. Massachusetts Comprehensive Assessment System: 14

15 Session 2, Open Response Question #28 28 Esther shot two arrows at a target. The picture below shows where the arrows landed. Esther calculated her score by adding the number of points for each ring in which an arrow landed. For the two arrows above, her score was 35 points ( ). a. In your Student Answer Booklet, make a list of all the possible scores Esther could have gotten by shooting two arrows that hit the target. b. Is it possible for Esther to score a total of 235 points using only 5 arrows? Show your work or explain your answer. c. What is the fewest number of arrows required for Esther to score a total of 240 points? Show your work or explain your answer. Reporting Category for item 28: Number Sense and Operations Massachusetts Comprehensive Assessment System: 15

16 Question 28 Scoring Guide Score 4 3 Description The response shows a comprehensive understanding of how to select and use appropriate operations to solve problems. The response shows a general understanding of how to select and use appropriate operations to solve problems. 2 The response shows a basic understanding of operations. 1 The response shows a minimal understanding of operations. 0 The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No Response. Score Point 4 a b. Yes 10x x x X c x 100 x x x x x Massachusetts Comprehensive Assessment System: 16

17 Score Point 3 A arrow #1 arrow #2 arrow #1 arrow # There are 16 possible combinations Ester could have shot. 20, 50, 100, 200, 125, 75, 150, 110, 60, 35 are the sums that both arrows together could have equaled. B No, Ester could not score a total of 235 points using 5 arrows. Using 5 arrows there are no combinations or numbers that Ester could shoot that would equal 235. For example using 5 arrows ester could have a total of 220, or 250 but there are no number combinations that would equal 235. C The fewest number of arrows required for ester to get 240 points is 6 arrows. Ester could get 200 points with his first two arrows (100 points each) and then 10 points with each of the remaining four arrows. This equals 240 points. Massachusetts Comprehensive Assessment System: 17

18 Score Point 2 a) b) min Yes min ester could have got a , 25, and missed the other 2. c) The fewest would be arows to get a total of 10 3 of 14 points Score Point = = = = = = = = = 50 Massachusetts Comprehensive Assessment System: 18

19 Score Point #1 continued No, because if she uses 5 it will go over 235 but if she does less than 5 it will be under 235. the fewest number of arrows would be 5 because you need = 240. Score Point 0 a. She could had got x b. No because she would have had 370. c. It would be only 1 arrow. Massachusetts Comprehensive Assessment System: 19

20 Session 2, Open Response Question #29 29 Molly formed three polygons a triangle, a rectangle, and a pentagon with string. She calculated the sum of the measures of the interior angles for each polygon and entered her data in the chart shown below. Type of Polygons Number of Sides Sum of the Measures of the Interior Angles Triangle Rectangle Pentagon Hexagon 6? Octagon 8? Unnamed Polygon? 2340 n-sided Polygon n? a. What is the sum of the measures of the interior angles of a hexagon? b. What is the sum of the measures of the interior angles of an octagon? c. How many sides does an unnamed polygon have if the sum of the measures of the interior angles is 2340? d. Explain how you would find the sum of the measures of the interior angles of an n-sided polygon. Reporting Category for item 29: Geometry Massachusetts Comprehensive Assessment System: 20

21 Question 29 Scoring Guide Score Description The response shows a comprehensive understanding of the relationship between the number of sides and the sums of angle measures of polygons. The response shows a general understanding of the relationship between the number of sides and the sums of angle measures of polygons. The response shows a basic understanding of the relationship between the number of sides and the sums of angle measures of polygons. The response shows a minimal understanding of the relationship between the number of sides and the sums of angle measures of polygons. The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No Response. Score Point 4 (a) The sum of the measures of the interior angles of a hexagon are 720º. (b) The sum of the measures of the interior of an octagon are 1080º. (c) There are 15 sides in the un-named polygon whose interior angles add up to 2340º. (d) To find the measure of an n-sided polygon, I used the formula [(n-2)2] 90 2 is subtracted from n and the new number is multiplied by 90 to get the sum of the measures of all the interior angles. To figure out the numbers of sides, divide the measure of all the interior angles by n. Massachusetts Comprehensive Assessment System: 21

22 Score Point 3 A. The interior angle is 720º Due to you add 180º each time you change shapes. B. The sum of a octagon is 1080º. C. It MUST Have 15 sides to add up to 2340º. D. I would add 180º for each side on the pollygon. Score Point 2 A Inteor angles of a hexagon is 720º I found this by adding 180º to 540º because of the hexagon having 6 sides. B Interior angles of an octagon = 1080º. I found this by adding 360º to 720º. I did this because the sides of an octagon increased by 2.(8). C The sides of the unnamed polygon are 13 sides. The way I figured this out was by first multiplying 180º by 14. This gave me 2520º. I was close, but that answer was not right. So, I then multiplied 180 by 13. This was the correct number I needed. Massachusetts Comprehensive Assessment System: 22

23 Score Point #2 continued D The way I thought you could find the measure of an n-sided polygon was knowing that as many times the sides measure, you add, double, or even triple 180º and add it on to the number of angles for the shape of before. For example, a triangle had 3 sides so you start of f with 180º. Then a rectangle had four sides so you add 180 to that giving you 360º: Then you could go from a rectangle to a hexagon which has 3 more than a rectangle (6). By doing this you then multiply 6 by 180 giving you 720º. In conclusion, by using this method, you will be able to find the measures of the interior angles. Score Point 1 (a) Hexagon = 720º, (b) Octagon = 900; (c) 10 sides for the unamed polygon; (d) You would 360 to the unamed polygon to get the number of the sum of the measures of the interior angles. Score Point 0 (A) The sum of the measures of the Interior angle for a hexagon is 1080º because you would do 6 (for the 6 sides) x 180º which is equal to 1080º. (B) There is 1440º in a octagon because 8 (for sides) x 180º equal 1440º. (C) 13 sides in a unamed polygon because it is 2340º in it so you do = 13. (D) If you # of sides was 10 you would do 10 x 180 = 1800º. Massachusetts Comprehensive Assessment System: 23

24 Session 2, Open Response Question #39 39 Use the ruler included in your reference sheet to answer question 39. The figure shown above represents the base of a cylindrical tank. The tank has a height of 16 centimeters (1 milliliter = 1 cubic centimeter). a. What is the radius of the base, in centimeters? b. What is the volume of the cylinder in milliliters? Show your work. c. If both the radius and the height of the cylinder were doubled, what would be the volume of the cylinder in milliliters? Show your work. d. Based on your answers to parts b and c, what is the ratio of the volume of the smaller tank to the volume of the larger tank? Show your work. Reporting Category for item 39: Measurement Massachusetts Comprehensive Assessment System: 24

25 Question 39 Scoring Guide Score Description The response demonstrates a comprehensive understanding of how a change in one variable results in a change in another variable by correctly answering the four parts of the question. The response demonstrates a general understanding of how a change in one variable results in a change in another variable by correctly answering three parts of the question. The response demonstrates a partial understanding of how a change in one variable results in a change in another variable by correctly answering two parts of the question. The response demonstrates a minimal understanding of how a change in one variable results in a change in another variable by correctly answering one part of the question. The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No Response. a. r = 5cm b ,256 milliliters c ,048 milliliters Score Point 4 d ratio = Massachusetts Comprehensive Assessment System: 25

26 Score Point 3 A. the radius is 5 centimeters B. v = π r 2 h v = π 5 16 v = π x 25 x x 100 v = 79 x ,400 mililiters v = 1264 centimeters 2 C. v = π x 100 x 32 v = π x 100 = v = 10,053 cent. 1,005,800 mililiters D. 1,005,300 > 126, times L = π 2r 2 x 16 = 2 small = S large = L S8 = L Score Point 2 Problem 39 (a) 100 cm 2 (b) v = milliliters 3.14 x 100cm 2 x 16cm x 16 = r = 200 H = 32 v = x x 32 (c) ratio 3.21 Massachusetts Comprehensive Assessment System: 26

27 Score Point 1 A. The Radius of the base is 15 cm. I found this by measuring from the center of the circle to the wall which gave the Radius. B. The volume of the cylander is ml. This is how I got that answer: = 16 ml. C. If both the Radius and height were doubled the new volume would be ml. This is how I got my answer: = D. I have no Idea how to do a ratio. I have a vague Idea what one is, but I don t have a chance of doing this problem correctly if I don t know what a Ratio is or how to figure one out. Score Point 0 The radius of the base, in centimeters is 9.9. The volume of the cylinder in milliliters is 10. I rounded 9.9 to 10. If both the radius and the height of the cylinder were doubled, the volume of the cylinder in milliliters would be 42. I just added I added 16 twice because it said the height of the cylinder were doubled. Add which equals 32 to 10 because that is the volume of the cylinder in millimitors and you get 42. The ratio to the smaller tank to the volume of the larger tank would be 21. I devided 42 by 2 to get that because the smaller tank would be smaller so just devide by the larger tank to get the answer to the smaller tank and you get 21. Massachusetts Comprehensive Assessment System: 27

28 The Massachusetts Comprehensive Assessment System (MCAS) Release of 2001 Test Items Mathematics Grade 8 Massachusetts Comprehensive Assessment System: 28

29 Massachusetts Comprehensive Assessment System: 29

30 Session 1, Open Response Question #8 8 The pattern shown below is for a square prism. The lengths of the line segments in the pattern were chosen so that the pattern could be folded along the dotted lines into the prism shown. a. Make a sketch of a pattern for a triangular prism. Label each line segment with a length that will make it possible to fold the pattern into the triangular prism. b. Make a sketch of a pattern for a cylinder. Label each line segment and diameter in your pattern with a length that will make it possible to create the cylinder from the pattern. Reporting Category for item 8: Geometry Massachusetts Comprehensive Assessment System: 30

31 Question 8 Scoring Guide Score Description Student shows comprehensive sense of spatial relationships by making accurate sketches of patterns for two three-dimensional geometric figures and labeling the lengths of edges appropriately. Student shows good sense of spatial relationships by making accurate sketches of patterns for two three-dimensional geometric figures and labeling the lengths of edges. One or two significant measures may be omitted. Student shows partial sense of spatial relationships by inconsistently making sketches of patterns for two three-dimensional geometric figures or inconsistently labeling the lengths of edges. Student shows limited sense of spatial relationships by making major errors in sketches and labeling. Response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No response. Score Point 4 A. 1cm 1cm 1cm 1cm 1cm 1cm 3cm 3cm 3cm 3cm 1cm 1cm 1cm 1cm 1cm Massachusetts Comprehensive Assessment System: 31

32 Score Point #4 continued B. 2.87cm 9cm 8cm 2.87cm Score Point 3 a Massachusetts Comprehensive Assessment System: 32

33 Score Point #3 continued b. perimeter = perimeter = 1 Score Point 2 a. 2 1 b Massachusetts Comprehensive Assessment System: 33

34 Score Point 1 a b Massachusetts Comprehensive Assessment System: 34

35 Score Point 0 a) Massachusetts Comprehensive Assessment System: 35

36 Session 1, Open Response Question #12 12 An eighth-grade class will perform the first four acts in the annual talent show. Every student is in exactly one of the four acts. The order in which the acts will be presented is to be decided by a drawing so that each act has an equal chance of being drawn. a. Chantal is a member of the eighth-grade class. What is the probability that her act will be presented first? b. Chantal's act was chosen to be presented first. Make a tree diagram, chart, or list showing all the possible orders in which the other three acts could be presented. Use the letters A, B, and C to represent these three acts. c. Rory, Jesse, and Chantal are all members of the eighth-grade class who will each perform an act. What is the probability that Rory's act will immediately follow Jesse's? Explain how you found your answer. Reporting Category for item 12: Analysis, Statistics, and Probability Massachusetts Comprehensive Assessment System: 36

37 Question 12 Scoring Guide Score Description The response demonstrates comprehensive understanding of the concepts of probability and basic combinatorics by accurately describing outcomes and events and determining the probability of those events. The response demonstrates general understanding of the concepts of probability and basic combinatorics by describing outcomes and events and determining the probability of those events. The response demonstrates basic understanding of the concepts of probability and basic combinatorics by describing outcomes and events and/or determining the probability of events. The response demonstrates minimal understanding of the concepts of probability and basic combinatorics by describing outcomes and events and/or determining the probability of events. The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No response. Score Point 4 A. The probability that her act will be presented first is ¼ of a 25% chance. The ways the three other acts could be presented are: A, B, C A, C, B C, A, B B, A, C B, C, A C, B, A Massachusetts Comprehensive Assessment System: 37

38 Score Point #4 continued C. The possibility that Rory s act will immediately follow Jesse s act is 1/3. This is because there are six possible placement for the three acts following Chantel. Out of the six possibilities Rory is only right behind Jesse twice. This makes the possibility 2/6 which can be reduced to 1/3. Possible Placements J = Jesse O = Other C, S, R, O R = Rory C = Chantal C, J, O, R 2/6 or 1/3 C, R, J, O C, O, J, R C, O, R, J C, R, O, J Score Point 3 a) The probability that her act will be presented First is a 1 out of four chance. This is because there is four acts in total, and her act is only one. so, she has a 1 out of four chance, or a ¼ chance. b) 1. A, B, C 2. A, C, B 3. B, A, C 4. B, C, A 5. C, A, B 6. C, B, A c) There is a 1 out of 6 chance that Rory s act will immediately follow Jesse s, because there are a total of six orders they can go in and only one out of the six orders would mean Rory goes after Jesse. So, there is a 1 out of 6 chance. Massachusetts Comprehensive Assessment System: 38

39 Score Point 2 A. If there are four act and she s just one of the 8 th graders, the probability that her act will be presented first is 1 4. B. Possible order for next 3 acts C. AB AC BA BC CA CB Score Point 1 a) 1 4 b) Chantal Chantal Chantal A B C A C B C B A c) 50% chance. If Jesse s goes firs then there are only Rory & Chantal to compete next. Which is 50/50. Massachusetts Comprehensive Assessment System: 39

40 Score Point 0 Chantal s possibility that her act would get presented first is 1 out of 3. The other groups can have a better chance of their group getting performed that if Chantal s group was not done doing what they had to get ready to perform, that s how the other group could of got put in front of Chantal s group. The probability of Rory s group following Jesse s group 1 out of 2, because if Chantal s group is done her group might go befor Jesse but this probability is 1 out of 2. Unless Jesse s group is nervice and don t want to go right after Rory s ground ant the that s how Chantal s group will go after Rory s group. Massachusetts Comprehensive Assessment System: 40

41 Session 2, Open Response Question #23 23 An eighth-grade class took a survey and found that the most popular types of music in their school were alternative rock, rap, and classic rock. They took a second survey to find out the students' preference among these three types of music. These are the results for 120 students. Favorite Types of Music Alternative Rock Rap Classic Rock a. Make a rough sketch of a circle graph displaying these data. Tell how many degrees should be in each sector of the graph. b. Explain how you find the number of degrees for each sector. Reporting Category for item 23: Data Analysis, Statistics, and Probability Massachusetts Comprehensive Assessment System: 41

42 Question 23 Scoring Guide Score Description Student makes reasonable sketch of circle graph and specifies correct number of degrees for each sector. Explanation or work is clear and shows correct strategy. Student makes reasonable sketch. Explanation or work shows correct strategy for determining number of degrees. Errors are clearly careless. OR Student makes reasonable sketch and specifies correct number of degrees. Explanation is vague but indicates correct strategy. Student makes reasonable sketch and gives correct percent for each sector, but no degrees. Explanation may or may not be included. OR Student specifies correct number of degrees for each sector. Explanation is incomplete, hard to follow, or missing. Sketch may or may not be included. OR Student's explanation shows understanding of strategy that can be used for determining number of degrees in a sector other than the 180 sector for alternative rock. Sketch may or may not be included. OR Student makes reasonable sketch and gives clear explanation based on fractional parts of circle. Answer has no conceptual errors regarding percents or degrees. Student's sketch of graph shows some understanding of data. OR Response shows a minimal understanding of problem. Response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No response. Massachusetts Comprehensive Assessment System: 42

43 Score Point 4 Rap 33% Classical Rock 17% 180º Alternative Rock R 50% = x x = = x x = 60 boys who like how many Cross multiply alternative rock degrees? 360 x = 180º = x 360 total number of boys degrees in a circle = x 360 = x x = 120º 360 x = 60º Massachusetts Comprehensive Assessment System: 43

44 Score Point 3 Classical Rock 60º Alternative Rock Rap 180º 120º First, I divided the amount of boys that like Alternative Rock into 120, and saw that it was half of 120. So I then divided that into 360º, because that is the amount of degrees in a circle, and saw that there should be 180º for alternative rock. I then went back and did the same thing for classical rock and rap. Score Point 2 50% You take the number of 33% boys and put them into a fraction so say alternative rock you were trying to figure out you would put 60 over % Massachusetts Comprehensive Assessment System: 44

45 Score Point 1 A. Alternative Rock 50º Classic RAP Rock 30º 20º B. B. I divided Score Point 0 40 Rap 20 Classic Rock 60 Alternative Rock Alternative rock is the most favorite in the school so it gets the most votes then comes rap then classic rock. Massachusetts Comprehensive Assessment System: 45

46 Session 3, Open Response Question #38 38 Ms. McCarthy's class is making up number puzzles. These are two of the puzzles. Manuel's puzzle: My number is even. It is a factor of 198 and a multiple of 9. It is less than 100. What is my number? Haan's puzzle: My number is the product of three different prime numbers. It is an odd number less than 125. The sum of its digits is a multiple of 3. One of its factors is the third prime number. What is my number? a. What is Manuel's number? b. What is Haan's number? Explain the strategy you used to find your answer to Haan's puzzle. c. Write a number puzzle that has exactly three clues, has one and only one answer, and includes the following words: factor and prime number. Reporting Category for item 38: Number Sense and Operations. Massachusetts Comprehensive Assessment System: 46

47 Question 38 Scoring Guide Score Description The response demonstrates a thorough understanding of number theory concepts of prime numbers, factors, and multiples and their use in solving problems. The response demonstrates a general understanding of number theory concepts of prime numbers, factors, and multiples and their use in solving problems. The response demonstrates a partial understanding of number theory concepts of prime numbers, factors, and multiples and their use in solving problems. The response demonstrates a minimal understanding of number theory concepts of prime numbers, factors, and multiples and their use in solving problems. The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No response. Score Point 4 a. Manuel s number is 18 because 18 is even, is a factor of 198 ( = 11), is a multiple of 9(9 x 2 = 18), and is less than 100. b. Haan s number is 105 because it is a product of 3 different primes (3 x 5 x 7 = 105), an odd number less than 125, the sum of the digits is a multiple of 3 ( = 6, 3 x 2 = 6), and one of the digits is the third prime factor. I got the answer by process of elimination and trial and error. I knew that one of the primes couldn t be 2 (that would make it an even number), so the one I tried was 3 x 5 x 7 and it worked. c. My number is odd between 0 and 30. My number is NOT prime. My number has 7 as a factor. What is my number? Answer: 21 Massachusetts Comprehensive Assessment System: 47

48 Score Point a. n = even n x = 198 n 9 : n< = = even = = even 56 x = = 6 answer: 56 b. y y z=n x, y, z = prime n = odd n<125 1x5x3=15 n = = = = x7x3= = = =2 5 = 21 5x7x3=105 5x7x3=105 answer: 105 I used the strategy of multiplying the first 3 odd prime numbers to get 105, then checked it to see if it worked for the rest of the rules. c. My number is a prime number. H is a factor of 121 H is less than 50 answer: 11 Massachusetts Comprehensive Assessment System: 48

49 Score Point 2 a) 18 b) 15, I figured the third prime number is 5. I then thought of two other prime numbers that were lower than 5. I got 3 and 1. 5 x 3 x 1 = 15 c) My number is not even. It is a factor of 26. It is a prime number. The answer is 13. Score Point 1 a) Manuel s number is 22. b) Haan s number is 15. I took the first three prime numbers and found the answer. Guess and check method. c) even number sum of first two prime numbers has 1 pair of factors that are the same answer = 4 Score Point 0 a) b) c) my number is even it is less then 20 anything multiplied by it is doubled what is my number my number is a prime number Massachusetts Comprehensive Assessment System: 49

50 Session 3, Open Response Question #39 39 Some eighth-grade students want to raise at least $300 for a field trip by selling popcorn and fruit bars. The chart below shows the amount of profit they will make on each sale. Profit from Sales Box of popcorn 60 Fruit bar 30 a. If they sell exactly 500 fruit bars, how many boxes of popcorn will they need to sell to make a total of $300? b. On the grid in your Student Answer Booklet, draw a graph showing the combinations of boxes of popcorn and fruit bars they must sell to make a total of exactly $300. Let the horizontal axis represent the number of fruit bars. Label that axis to 1,000. Show or describe the calculations you used to find the data points for your graph. c. Based on last year's sales, the students will probably not be able to sell more than 600 fruit bars. Using your graph, explain how you can find the number of boxes of popcorn the students must sell to make a total of $300 if they sell exactly 600 fruit bars. How many boxes of popcorn must they sell? Reporting Category for item 39: Patterns, Relations, and Algebra. Massachusetts Comprehensive Assessment System: 50

51 Question 39 Scoring Guide Score Description Student demonstrates comprehensive understanding of functional relationships by representing a real-life situation accurately in an appropriate graph, explaining the strategy used, and using the graph to find related values. Student demonstrates general understanding of functional relationships by representing a real-life situation in an appropriate graph, explaining the strategy used, and using the graph to find related values with minor inaccuracies. Student shows basic understanding of functional relationships by representing a real-life situation in an incomplete graph, ineffectively explaining the strategy used, and/or failing to use the graph to find related values. Student demonstrates minimal understanding of the relationship between the variables or of graphing. Response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No response. Score Point 4 A. Fruit = 150. They will need to sell 250 boxes of popcorn. B. See the grid on the next page. C. In order to find out how many boxes of popcorn they must sell you would refer to the graph. You would look on the x-axis for 600 which represents the 600 fruit bars. Then follow up that line until you reach the point. Then trace that back to the y-axis which represents the popcorn. It will read 200. You now know that you will need 200 popcorn boxes sold for a total of 120 and when added to the 600 fruit bars total of 180 you get 300 dollars. Massachusetts Comprehensive Assessment System: 51

52 Score Point #4B continued B. x Fruit Bars & Popcorn sales (in hundreds) # of popcorn boxes y Score Point 3 a) There is an expression you can use to solve this:.60p+.80f = p=popcorn & f=fruit.60p+.30(500) = 300 a:.60p-150 *they will need to p = 250 sell 250 boxes of popcorn to have #300 b) See the graph on the next page. c) My graph says that the students would have to sell 200 boxes of popcorn if they sell exactly 600 fruit bars to make a total of $300. Massachusetts Comprehensive Assessment System: 52

53 Score Point #3B continued y Fruit Bars Popcorn x y x I used the equation.60p +.30f = 300 Score Point 2 (A). I they sold 500 Fruit Bars at 0.30 cents they would have made $ dollars. I got this answer by multiplying 500 fruit bars by 0.30 cents and I got 250. So they would have to sell 250 bars of popcorn to make the goal of dollars. (C). If the students sold 600 Fruit bars they would make $ dollars. The students would have to sell exactly 200 popcorn bags, to make dollars. Massachusetts Comprehensive Assessment System: 53

54 Score Point #2 continued FB PC Score Point 1 A. They will need 250 boxes of Popcorn. B. I will need twice as many popcorn as fruitbars. C. I would half to put the right amount of money on the graph Fruit Bars Massachusetts Comprehensive Assessment System: 54

55 Score Point 0 A. / They would have to sell 18 boxes of popcorn. B. / 60 popcorn Fruit bars C. / 18 boxes of popcorn and 600 fruit bars. Massachusetts Comprehensive Assessment System: 55

56 The Massachusetts Comprehensive Assessment System (MCAS) Release of 2000 Test Items Mathematics Grade 8 Massachusetts Comprehensive Assessment System: 56

57 Session 1, Open Response Question #8 8 John is playing a board game that uses a pair of number cubes with sides numbered 1 to 6. To find how many spaces he can move on the board, he adds the two numbers he rolls. The possible sums are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. a. Are all the sums John can roll equally likely? Explain your reasoning in detail. b. John needs to roll a sum of exactly 11 in order to get another turn. What is the probability that he will roll a sum of exactly 11? Explain your reasoning in detail. Reporting Category Substrand for item 8: Statistics and Probability; Probability. Massachusetts Comprehensive Assessment System: 57

58 Question 8 Scoring Guide Score Description The response demonstrates comprehensive understanding of the concepts of theoretical probability by accurately describing outcomes and events and determining the likelihood and probability of those events. The response demonstrates general understanding of the concepts of theoretical probability by describing outcomes and events and their probability. Any errors are minor. The response demonstrates basic understanding of some concepts of theoretical probability. The response shows minimal understanding of some concepts of theoretical probability. The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No response. Massachusetts Comprehensive Assessment System: 58

59 Score Point 4 a. All the sums that John can role out are not equally likely to happen. This occurs because some sums have more numbers than others that could be added to receive that sum. Here is an example: 1+1=2 2+1=3 3+1=4 4+1=5 5+1=6 6+1=7 1+2=3 2+2=4 3+2=5 4+2=6 5+2=7 6+2=8 1+3=4 2+3=5 3+3=6 4+3=7 5+3=8 6+3=9 1+4=5 2+4=6 3+4=7 4+4=8 5+4=9 6+4=10 1+5=6 2+5=7 3+5=8 4+5=9 5+5=10 6+5=11 b. The probability that John will role an eleven is I found this answer by first finding two dice. It was 6x6=36. Now there were two ways to get an eleven. A 5 on the first dice and 6 on second dice or 6 on first dice and 5 on second dice. You get 2 36 or As you can see the 7 occurs all 6 times while other numbers occur less. 2 occurs only once 3 occurs twice 4 occurs three times 8 occurs five times and so on Score Point 3 A. Not all of the sums John can roll are equally likely. Seven is the most likely to get because there are six different possibilities of rolling a seven. B. John has two chances of rolling an eleven he rolls a five or a five and a six. (See next page for details.) Massachusetts Comprehensive Assessment System: 59

60 Score Point #4B continued , , 2+2, , 2+3, 3+2, , 2+4, 3+3, 4+2, , 2+5, 3+4, 4+3, 5+2, , 3+5, 4+4, 5+3, , 4+5, 5+4, , 5+5, , Score Point 2 a. Yes because there is equal probability. b because there are 36 sums and 2 equil 11. Score Point 1 A. his sums are all equal because if each number cube adds up to six then you cant get higher than a 12. B. the probability that he will roll an 11 is 1 out of 36 because you can only get that number with a 6 and a 5 and there are 36 combinations with 2 number cubes so, there is a 1 out of 36of chance.. Score Point 0 a. yes all sums ar equally likely to be rolled. This is because no matter what two numbers he rolls he will get one of these sums. b. His protability to roll an 11 is 1 out of 6. The only two numbers, he can get, to get 11 are 5 and 6 out of a total of 6 different numbers. Massachusetts Comprehensive Assessment System: 60

61 Session 1, Open Response Question #12 12 Erin is writing a science fiction story. She has invented a money system for her planet that uses four coins that she drew and named like this: She has challenged her classmates to determine the relationships among the values of the coins from the following clues. a. Use Clue 1 above to find how many s equal 1. Use words or pictures to explain your reasoning. b. Use Clue 2 to find out how many s equal 1. Use words or pictures to explain your reasoning. c. Use Clue 3 and your answers to parts a and b to find how many s equal 1. Use words or pictures to explain your reasoning. d. Erin told her classmates that 1 is worth 25 in U.S. money. What is the value in U.S. money of each of the following? Reporting Category Substrand for item 12: Patterns, Relations, and Functions/Algebra. Massachusetts Comprehensive Assessment System: 61

62 Question 12 Scoring Guide Score Description The response demonstrates thorough understanding of models of simple linear situations by correctly solving equations represented by models, clearly explaining the process, and evaluating the expressions for given values. The response demonstrates general understanding of models of simple linear situations by solving equations represented by models, explaining the process, and evaluating the expressions for given values with only minor errors. The response shows basic understanding of the concept of simple linear situations by solving some equations and/or evaluating some expressions for given values. 1 The response shows minimal understanding of simple linear situations. 0 The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No response. Score Point 4 a. 2 s = 1. I started with 3 = 1 + 1, then subtracted 1 from end equasion. b. 3 s = 1. I set up the equasion as + 2 = 2 1 First, I added to site of the equasion, leaving + 3 = 2. Then I subtracted 1 from each side to get 3. Massachusetts Comprehensive Assessment System: 62

63 Score Point #4C continued c. 4 s = 1. I set up the equasion as = First, I reconciled the 3 because I knew from part b. that they are equal, leaving I knew from 2 = 1 and therefore that * = 2, so I subtracted the 4 and 2 leaving 1 = 4. d. 1 = 50 1 = $ = $6.00 Score Point 3 a. 3 s are equal to 1. This is stated in clue 1. 2 c s - 1T ccc=tc = b. I did guess and check with H=9 and R= 3. This worked. H =15 1H=3R s 18:3 H>R c. 1R=3T 2=T R>T RCH 1H=3R 3RT 1R+4C+T4 = 3R+6T 1R+2C+6T 1R+1C+5T 2C-1T 1T=3R 1R=3T d. C =.25 T =.50 R = $1.50 H = $ $ $4.50 Massachusetts Comprehensive Assessment System: 63

64 Score Point 2 A. = & = & -- two are in You can automatically cross out two s from the equation because they have the same value. = If two = four s then one = two s B. s are less than 3 C. Game costs Paid with two s Three s in an D. = 25 = 50 = 75 = $2.25 Score Point 1 a. 2 s are equal to 1 b. 2 s = 1 c. Score Point 0 a) s are equal to 1 b) a 1 2 equals to 1 c) 2 s equa1 1R d) 1 = = 50 1 = 10 Massachusetts Comprehensive Assessment System: 64

65 Session 2, Open Response Question #23 Use your ruler to answer this open-response question. Jarrod is the editor of the school newspaper. In the next issue, a page will be devoted to a list of the students who 23 perform community service. Jarrod is planning how to arrange the names. The first figure below tells the size of the page and the headline. The second figure shows the actual size of type that will be used for the students' names and the actual width of each column. There are 175 students who performed community service. Jarrod wants to plan the page so that the page has the greatest number of columns possible, and the columns are as close to the same length as possible. a. What is the greatest number of columns that Jarrod can put on the page? Show or describe how you found your answer. b. How many names should he put in each column so that the columns are of equal length or as close to equal length as possible? Assume each name will fit on one line in a column. c. How long will each column of names be? Show or describe how you found your answer. Reporting Category Substrand for item 23: Geometry and Measurement/Measurement. Massachusetts Comprehensive Assessment System: 65

66 Question 23 Scoring Guide Score Description The response demonstrates comprehensive understanding of the concept of measurement by solving a real-life problem involving the selection of an appropriate measurement tool and a correct strategy for making the measurement. The response demonstrates a practical understanding of the concept of measurement by solving a real-life problem involving the selection of an appropriate measurement tool and a correct strategy for making the measurement with only one or two computational errors. The response shows partial understanding of the concept of measurement by attempting to solve a real-life problem by selecting an appropriate measurement tool but not using a correct strategy for making the measurement. The response shows minimal understanding of the concept of measurement. The response is incorrect or contains some correct work that is irrelevant to the skill or concept being measured. Blank No response. Score Point 4 a. The greatest number of columns Jarrod can put on the page is 5. I figured this out by measuring the column width: 1½ in, and the page s width: 8½ in. I then divided 1.5 into 8.5 : 5.6 and since you can t have half a column, I rounded it down to 5. b. He should put 35 names in each column. To figure that out, I simply divided 5 (#of columns) into 175 (#of names). c. The columns will be 6.5 inches long. I figured that out by measuring how long the 7 names they gave us together were and they were about 1.3 inches long. There were 7 names and 7 goes into 35 5 times, so I then just multiplied 1 x 3 x 5 and got 6.5. Massachusetts Comprehensive Assessment System: 66

67 Score Point 3 A. The greatest number of columns would be 8½ 1½ (The smallest column size). So, Jarrod can fit 5 because 1 inch is left over. this can be used for a half inch border. B. He should put 35 names in each row because 175 names 5 columns = 35 names names column. C. Well, Since each text name is 1 8 tall and there are 1 16 inch squares between names with a 1 8 inch top and bottom border. 35 names * 1 8 inch = 4 inches top & 1 bottom border = 2 8 inch 34 spaces * 1 16 = 21 8 inch = Total Column 58 8 = 7.25 or Each column is inches long. Massachusetts Comprehensive Assessment System: 67

68 Score Point 2 a) The greatest amount of columns that Jarrod can put on his page is five, because if each column is 1½ inches wide, only five columns can fit into his 8 ½ inches wide page. This was figured out by doing 1.5*5 which is 7.5. b) There should be 35 names per column, because 5 goes into 175 evenly. c) Each column will be 9 inches long, because if his heading takes up two inches at the top of the page, a 9 inch column would fit the rest of the page. Score Point 1 6 names = 1 ½ width 1 1/2 long a. 8½ 1½ = columns b. 6 x 6 = 36 names in each column c. 11 inches 2 inches = 9 inches you need to subtract 2 for the heading. Massachusetts Comprehensive Assessment System: 68

69 Score Point 0 a. The greatest number of colums that you could put on this page is 2 colums. b. I think that he can put 87 on one side and 88 on the other. Because if you do 8½ x 11 you get 89½ but since you soud to be as accurate as possible so I would put 88 on one side and 87 on the other. c. I think each column will be five inches because it was eleven but you need to subtract the two inches for the heading and you get nine. Massachusetts Comprehensive Assessment System: 69

70 Session 3, Open Response Question #38 38 The planning committee at Lane Middle School is planning a pizza party for its 127 eighth-grade students. They got this menu from The Pizza Palace. The planning committee took a survey of a random sample of 26 eighth-grade students by asking, "What kind of pizza do you want?" This is what they found. Favorite Kind of Pizza Kind of Pizza Cheese Sausage Pepperoni Vegetarian Number of students The committee has a budget of $300 for the pizza. What kinds and sizes of pizzas could the committee order so that each of the 127 students can have his or her favorite kind of pizza? a. Explain how you used the results of the survey to decide which pizzas to order. b. Show or describe the calculations needed to be sure that there will be enough pizza for the 127 students. c. Show or describe the calculations needed to be sure that the cost of the pizza totals $300 or less. You do not need to find the cheapest way to buy enough pizza. You only need to make sure that the total cost is $300 or less. Reporting Category Substrand for item 38: Number Sense/Computation and Estimation. Massachusetts Comprehensive Assessment System: 70